Transcript Statistics - The Pingry School

Chapter 3
Averages and Variations
3.1 Measures of Central
Tendency
Mode, Median and Mean
What kind of data will we be able to
compute mode, median and mean?
Quantitative data can have a mode,
median and mean.
Qualitative data can have a mode.
Mode
The value that occurs most frequently is
the mode. Some books describe the
mode as the “hump” or local high
point in a histogram, which does imply
frequency of an answer.
Median
The median of a data set is the middle
data value.
To find, order the data from smallest to
largest, and the data set in the middle
(for a data set of n, the middle position
is n  1 ) is the median.
2
Does anyone detect a potential problem?
Mean
You are used to an “average” of the test. The
technical term is the mean.
Mean
You are used to an “average” of the test. The
technical term is the mean.
sum of data values
mean 
number of data values
Trimmed mean is a term for a mean where a
percentage of the data values are
disregarded. A 5% mean is one where 5% of
top and 5% of bottom values are thrown out
before computing the mean.
Pulse Data
Lets find the mode, the median and the
mean of the pulse data from the first
day of class.
We just found the population mean (μ)
rather than the sample mean ().
What is the difference then between μ
and ?
Weighted Averages
Final Exams are computed in as
weighted averages. How do they do
that???
Weighted Averages
Final Exams are computed in as
weighted averages. How do they do
that???
weighted average 


xw
w
That is, multiply the data value by its weighting, add each of those, then
divide by the sum of the weighting (typically 1)
3.2
Measures of Variation
While knowing the mean is
important
There is other information from data that
you can measure.
These tell you about the spread of the
data.
Range – difference between largest and
smallest value of a data distribution.
Variance
Variance = measure of how data tends to
spread around an expected value (the
mean)
Each data point = x
Mean = 
Deviation = x – 
Sample size = n
Variance = s2
Standard Deviation = s
Variance (cont)
(x  x)


2
s
2
n 1
Defining Formula
Variance (cont)
(x  x)


2
s
2
2
s 
Defining Formula
n 1
x
2
x



n 1
n
2
Computation Formula
Variance (cont)
To find standard deviation, just square
root the variance.
The computational formula tends to be
a little easier to do by hand, but we will
practice both.
These two formulas ARE the same.
Variance (cont)
Lets find the variance and the standard
deviation of the pulse data, using both
formulas.
Variance (cont)
If an entire population is used, instead of a
sample, the notation is different but the
methods are the same
Each data point = x
Mean = µ
Deviation = x – µ
Sample size = N
Variance = σ 2
Standard Deviation = σ
Variance (cont)
(x  )


2

2
N
Defining Formula
Variance (cont)
Coefficient of Variance (CV) expresses
standard deviation as a percentage of
the sample/population mean.
Variance (cont)
Coefficient of Variance (CV) expresses
standard deviation as a percentage of
the sample/population mean.
s
CV  100
x
Sample

CV  100

Population
Variance (cont)
Chebyshev’s Theorem
For any data set, the proportion that lies within
k standard deviations on either side of the
mean is at least
1
1 2
k
So 75% lies between 2 standard deviations,
88.9% between 3 standard deviations, etc.
3.3 Mean/Standard
Deviation
What if you use grouped data
Grouped Data
Lots of data = TEDIOUS, whether you
have a calculator or not… If you
generally approximate the mean and
standard deviation, that sometimes is
enough
To deal with this, you actually begin with
a frequency table (remember
Histograms?
Grouped Data (cont)
1.
2.
3.
4.
Make a frequency table
Find the midpoint of each class = x
Compute each class frequency = f
Total number of entries = n
Grouped Data (cont)
1.
2.
3.
4.
Make a frequency table
Find the midpoint of each class = x
Compute each class frequency = f
Total number of entries = n
xf

average  x 
n
Grouped Data (cont)
(x  x) f


2
s
2
2
s 
n 1

Defining Formula
xf 


x f
2
2
n 1
n
Computation Formula
Grouped Data (cont)
Essentially, by using the midpoint and the
frequency, you use a representation for ALL
data values in that class, without typing in
every data value.
It will be a little off, but again, if the data set is
huge it isn’t a bad way to approach the
problem.
3.4 Percentiles
Box/Whiskers Plots
Percentiles
Baby Calculator
Children’s BMI
A percentile ranking allows one to know
where the particular data value falls in
relation to the entire population.
Percentiles (cont)
The Pth percentile (1 ≤ P ≤ 99) is a value so that
P% of the data falls at or below it (and 100 –
P % falls at/above)
60th Percentile does NOT mean 60% score – it
means that 60% of scores fall at or below
that position… 60th percentile could be 80%
Where have you seen percentiles?
Percentiles (cont)
Quartiles – special percentiles used frequently.
The data is divided into fourths, called
Quartiles.
2nd Quartile – Median
1st Quartile – Median below (exclude Q2)
3rd Quartile – Median above (exclude Q2)
Interquartile Range (IQR) = Q3 – Q1
Percentiles (cont)
Lets find the quartiles for following Math
class sizes in the 9th grade.
10, 11, 12, 12, 14, 15, 16, 17, 19, 20
1st Q = 12
3rd Q = 17
Median = 14.5
IQR = 17 – 12 = 5
Percentiles (cont)
Lets find the quartile for the pulse data
Why are these values significant? These
are needed to make Box and Whiskers
Plots
Box and Whiskers Plots
Box and Whiskers Plots (cont)
The five number summary
is used to make a box and whisker plot.
Lowest value, Q1, Median, Q3, Highest Value
Lets make a box and whiskers plot for
the class size data.
20
Highest Value
18
Q2
16
Median
14
12
Q1
10
Lowest Value
Box and Whiskers Plots (cont)
Lets make a box and whiskers for the
pulse data
Outliers – data > Q3 + 1.5 IQR
data < Q1 – 1.5 IQR
Resources
• http://www.statcan.ca/english/edu/power/ch12/plots.htm
• http://www.statsdirect.com/help/graphics/box_whisker.htm
• http://v8doc.sas.com/sashtml/stat/chap18/sect18.htm